Properties of convergence of a class of iterative processes generated by sequences of self-mappings with applications to switched dynamic systems

This article investigates the convergence properties of iterative processes involving sequences of self-mappings of metric or Banach spaces. Such sequences are built from a set of primary self-mappings which are either expansive or non-expansive self-mappings and some of the non-expansive ones can be contractive including the case of strict contractions. The sequences are built subject to switching laws which select each active self-mapping on a certain activation interval in such a way that essential properties of boundedness and convergence of distances and iterated sequences are guaranteed. Applications to the important problem of stability of dynamic switched systems are also given.The authors are very grateful to the Spanish Government for Grant DPI2012-30651 and to the Basque Government and UPV/EHU for Grants IT378-10, SAIOTEK S-PE13UN039 and UFI 2011/07. The authors are also grateful to the referees for their suggestions

Convergence properties and fixed points of two general iterative schemes with composed maps in banach spaces with applications to guaranteed global stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems

Synchronization Problems in Networks of Nonlinear Agents

Over the last years, consensus and synchronization problems have been a popular topic in the systems and control community. This interest is motivated by the fact that, in several fields of application, a certain number of agents is interacting or has to cooperate to achieve a certain task. Robotic swarms, sensor networks, power networks, biological networks are only few outstanding examples where networks of agents displays behaviors which can be modeled and studied by means of consensus and synchronisation techniques. In this thesis we consider a general class of networked nonlinear systems in different operating frameworks and design control architecture to force the systems to reach synchronization and consensus on a target behavior. In particular, we consider the case of homogeneous and heterogeneous nonlinear agents with a static communication topology and design a static high-gain-based diffusive coupling and an internal model-based regulator respectively, to solve the problem of consensus. Then, we analyze the case of dynamical links and show under which conditions, synchronization for homogeneous nonlinear systems can be achieved. Depending on the structure of the dynamic links at hand, static and dynamic regulators (based on the concept extended state observers) are proposed. Furthermore, we address the problem of disconnected topology and switching topology and derive under which conditions agents reach cluster synchronization and synchronization respectively. Last, we consider the problem of a sampled exchange of information between the agents and design a triggering rule locally at each agent such that the overall network reaches synchronization

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

Multiscale derivation, analysis and simulation of collective dynamics models: geometrical aspects and applications

This thesis is a contribution to the study of swarming phenomena from the point of view of mathematical kinetic theory. This multiscale approach starts from stochastic individual based (or particle) models and aims at the derivation of partial differential equation models on statistical quantities when the number of particles tends to infinity. This latter class of models is better suited for mathematical analysis in order to reveal and explain large-scale emerging phenomena observed in various biological systems such as flocks of birds or swarms of bacteria. Within this objective, a large part of this thesis is dedicated to the study of a body-attitude coordination model and, through this example, of the influence of geometry on self-organisation. The first part of the thesis deals with the rigorous derivation of partial differential equation models from particle systems with mean-field interactions. After a review of the literature, in particular on the notion of propagation of chaos, a rigorous convergence result is proved for a large class of geometrically enriched piecewise deterministic particle models towards local BGK-type equations. In addition, the method developed is applied to the design and analysis of a new particle-based algorithm for sampling. This first part also addresses the question of the efficient simulation of particle systems using recent GPU routines. The second part of the thesis is devoted to kinetic and fluid models for body-oriented particles. The kinetic model is rigorously derived as the mean-field limit of a particle system. In the spatially homogeneous case, a phase transition phenomenon is investigated which discriminates, depending on the parameters of the model, between a “disordered” dynamics and a self-organised “ordered” dynamics. The fluid (or macroscopic) model was derived as the hydrodynamic limit of the kinetic model a few years ago by Degond et al. The analytical and numerical study of this model reveal the existence of new self-organised phenomena which are confirmed and quantified using particle simulations. Finally a generalisation of this model in arbitrary dimension is presented.Open Acces

Effective Non-Hermiticity and Topology in Markovian Quadratic Bosonic Dynamics

Recently, there has been an explosion of interest in re-imagining many-body quantum phenomena beyond equilibrium. One such effort has extended the symmetry-protected topological (SPT) phase classification of non-interacting fermions to driven and dissipative settings, uncovering novel topological phenomena that are not known to exist in equilibrium which may have wide-ranging applications in quantum science. Similar physics in non-interacting bosonic systems has remained elusive. Even at equilibrium, an effective non-Hermiticity intrinsic to bosonic Hamiltonians poses theoretical challenges. While this non-Hermiticity has been acknowledged, its implications have not been explored in-depth. Beyond this dynamical peculiarity, major roadblocks have arisen in the search for SPT physics in non-interacting bosonic systems, calling for a much needed paradigm shift beyond equilibrium. The research program undertaken in this thesis provides a systematic investigation of effective non-Hermiticity in non-interacting bosonic Hamiltonians and establishes the extent to which one must move beyond equilibrium to uncover SPT-like bosonic physics. Beginning in the closed-system setting, whereby systems are modeled by quadratic Hamiltonians, we classify the types of dynamical instabilities effective non-Hermiticity engenders. While these flavors of instability are distinguished by the algebraic behavior of normal modes, they can be unified under the umbrella of spontaneous generalized parity-time symmetry-breaking. By harnessing tools from Krein stability theory, a numerical indicator of dynamical stability phase transitions is also introduced. Throughout, the role played by non-Hermiticity in dynamically stable systems is scrutinized, resulting in the discovery of a Hermiticity-restoring duality transformation. Building on the preceding analysis, we take the necessary plunge into open bosonic systems undergoing Markovian dissipation, modeled by quadratic (Gaussian) Lindblad master equations. The first finding is that of a uniquely-bosonic notion of dynamical metastability, whereby asymptotically stable dynamics are preempted by a regime of transient amplification. Incorporating non-trivial topological invariants leads to the notion of topological metastability which, remarkably, features tight bosonic analogues to the edge modes characteristic of fermionic SPT phases - which we deem Majorana and Dirac bosons - along with a manifold of long-lived quasi-steady states. Implications regarding the breakdown of Noether\u27s theorem are explored, and several observable signatures based on two-time correlation functions and power spectra are proposed